Finite Algebras on the Semantic Web

A Senior Project submitted to

The Division of Science, Mathematics, and Computing

Bard College

Feifan Zheng

Annandale-on-Hudson, New York

May, 2013

Abstract

This project explores information sharing in mathematical research. Currently, the major

approaches to dissemination are journal articles, reviews, and graduate-level textbooks.

Other less formal approaches include conference talks, interpersonal conversations, math-

ematical software, and static websites.

Mathematical knowledge management has as its aim to improve the sharing of mathe-

matical research artifacts through the development of new data formats and other digital

means. In this work, we consider the problem of publishing the products of research in

universal algebra, speciﬁcally ﬁnite algebras and their properties, on the Semantic Web.

Contents

Abstract 1

Dedication 6

Acknowledgments 7

1 Introduction 8

1.1 Current Status of Information Sharing . . . . . . . . . . . . . . . . . . . . . 8

1.1.1 Journals, Reviews, and Textbooks[13] . . . . . . . . . . . . . . . . . 8

1.1.2 Conference Talks and Other Interpersonal Communication . . . . . . 9

1.1.3 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.4 Website Repositories . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 New Ways of Sharing Mathematics Information . . . . . . . . . . . . . . . . 10

1.3 The Solution – the Semantic Web . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.1 The Traditional World Wide Web . . . . . . . . . . . . . . . . . . . 12

1.3.2 The Modern World Wide Web . . . . . . . . . . . . . . . . . . . . . 13

1.3.3 The Semantic Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Triples and the Semantic Web 16

2.1 Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Representing Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1 Resource Description Framework (RDF) . . . . . . . . . . . . . . . . 22

2.2.2 Terse RDF Triple Language (Turtle) . . . . . . . . . . . . . . . . . . 23

3 Quandles 27

3.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1 General Deﬁnitions and Axioms . . . . . . . . . . . . . . . . . . . . 27

Contents 3

3.1.2 Example of Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Quandles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.1 Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.3 Results from the ASC Lab . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Quandles and Triples Transformation 32

4.1 Breaking Down A Quandle . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1.1 From operation table to “list of lists” Mathematica format . . . . . 32

4.1.2 From quandle to “list of lists” Mathematica format . . . . . . . . . . 37

4.2 Transformation Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 From Mathematica list of triple lists to RDF . . . . . . . . . . . . . 42

5 Conclusion 46

5.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Bibliography 49

List of Figures

1.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

LIST OF FIGURES 5

2.2.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Dedication

To my beloved parents.

献给我最挚爱的爸爸妈妈。

谁言寸草心，报得三春晖。

Acknowledgments

I would like to express my deep gratitude to Professor McGrail, my senior project advisor

and a great friend, for his continuous support, patient guidance, and enthusiastic encour-

agement.

I would also like to thank Professor Thomas and Professor Hsiao, my senior project board

members, for their useful critiques of this project.

Special thanks to my dear friends from Bard: Hsiao-Fang Lin, Xingye Zhang, Steven

Wu, Yu Qiu, Hanze Song, Lei Lu, Chi-Hui Yen, Winnie Yau, Xiaohao Wu;

And my friends from back home: 门门，狐狸，没没.

Finally, I wish to thank my parents for their undivided support and encouragement

throughout my study at Bard.

Introduction

1.1 Current Status of Information Sharing

Mathematical information for research purpose can be obtained through various sharing

approaches. However, there are plenty of research artifacts that are useful to mathemati-

cians but are not normally shared and veriﬁed. In another words, the sum total of the

sharing methods leaves out much important mathematical information.

1.1.1 Journals, Reviews, and Textbooks[13]

The primary formal approaches to acquire mathematical research results are through pub-

lished journal articles, reviews, and graduate-level textbooks, which traditionally appear

as hard copies. In modern days, these can also be downloaded from the internet. In gen-

eral, the content of a journal article only supports the main results of that work. Extensive

details and examples are frequently omitted from the work. Additionally, people must fol-

low the narrative in order to obtain the information from it. What are often missing,

or, at least, diﬃcult to extract are the countless examples in the course of mathematical

discovery.

1. INTRODUCTION 9

1.1.2 Conference Talks and Other Interpersonal Communication

Conference talks and interpersonal communication are also common approaches of sharing

mathematical information. These methods involves real-time interactions. The receiver of

information can pose further queries to the provider of information in real time. However,

the information shared through these approaches might lack accuracy. The facts presented

have not been peer-reviewed. Moreover, mistakes in communication might be more likely

in real time.

1.1.3 Software

The uses of computers in the mathematical research community are many. In the early

days of computing, mathematicians harnessed these machines for complex calculations.

Nowadays, computers are employed for mathematical typesetting, visualization, commu-

nication and the archive of digital journals. Moreover, there are a myriad of systems to

support mathematical research. We mention some of these below:

• “Canned” software

Many mathematicians make use of “canned” software in their research process. One

such example of a “canned” software is UACalc[14], the universal algebra calcu-

lator. UACalc allows users to input a ﬁnite algebra and determine whether that

algebra has certain properties. Another example is KnotPlot[15], which is a software

that visualizes and manipulates mathematical knots and braids in three and four

dimensions.

• Customizable, limited-scope software

Other mathematics researchers use many systems that are also for special circum-

stances that allow a limited degree of customizability. These systems allow users

to deﬁne their own tests, but the scope of those tests are limited. For example,

1. INTRODUCTION 10

the Mace4/Prover9[16] suite of ﬁrst-order utilities allows the development of custom

searches and veriﬁcations. However, those processes must be speciﬁed in ﬁrst-order

logic, which excludes most computable queries and tests.

• Full symbolic system

Mathematicians also use full symbolic systems such as Mathematica[3], Maple[18],

Matlab[19], and Wolfram Alpha[20] to design mathematical software. These are gen-

eral programing system that are optimized for use in a wide variety of mathematical

applications.

1.1.4 Website Repositories

Website repositories also contains a massive amount of mathematics information that

Mathematicians often require. Information shared through this method often covers com-

plex mathematics concepts and topics, and is maintained by experts of mathematics. These

website repositories can be maintained either privately or publicly. Website repositories

such as MathWorld[21] and Knot Atlas[22] are maintained by private organizations, while

for websites such as Wikipedia[23], information editing is open to the public. Both pri-

vately and publicly maintained website repositories have their own ﬂaws. For example,

information from publicly maintained websites is often subject to higher variation in reli-

ability, while mistakes that are found in private website repositories are hard for and users

to correct.

1.2 New Ways of Sharing Mathematics Information

• Mathematical Knowledge Management (MKM)

Mathematical Knowledge Management[10], also known as MKM, is an innovative

ﬁeld that focuses on developing new techniques to help in presenting, produc-

ing, transmitting, consuming, and managing sophisticated mathematical knowledge.

1. INTRODUCTION 11

MKM involves subjects such as mathematics, computer science, library science, and

scientiﬁc publishing. Areas of research in MKM include, but are not limited to:

– Repositories of formalized mathematics;

– Web presentation of mathematics; and

– Collaboration tools for mathematics.

• Digital Mathematics Library (DML)

Digital Mathematics Library[11], also known as DML, is a project that pursues a

global mathematical digital library. Topics of inquiry in DML include, but are not

limited to:

– Search, indexing, and retrieval of mathematical documents;

– Mathematical document compression; and

– Long term archiving, data migration.

• OpenMath

The OpenMath project[12] has the aim of formulating a standard for representing,

exchanging, storing, and publishing mathematical objects along with their semantics

on the World Wide Web. Topics pertinent to OpenMath include:

– Displaying mathematical objects in a browser;

– Exchanging mathematical objects between software systems; and

– “Cutting and pasting” mathematical objects for use in diﬀerent applications.

1.3 The Solution – the Semantic Web

The Semantic Web[7] is a project to create one giant, distributed, and linked database on

the World Wide Web. This will allow people to share information beyond the boundaries

1. INTRODUCTION 12

of applications and websites. In the Semantic Web, information is linked on a global scale

and can be easily processed by machines. In another words, the Semantic Web functions

as a globally-linked database that allows computers to seek out knowledge distributed

throughout the Web, compile it, and then take actions based on it.

The Semantic Web is a somewhat utopia vision of the World Wide Web. The ultimate

diﬀerence between the traditional web and the Semantic Web is described by Tim Berners-

Lee, the the inventor of the World Wide Web, as below:

“If HTML and the Web made all the online documents look like one huge book,

RDF, schema, and inference languages will make all the data in the world look

like one huge database”[2].

1.3.1 The Traditional World Wide Web

The original interaction model for the traditional World Wide Web, also referred as the

Web, proceeds as follows:

1. The web browser makes a request speciﬁed by a uniform resource locator (URL) to

the web server. The URL speciﬁes a server, a path on that server, and the document

of interest. See Figure 1.3.1.

Figure 1.3.1.

1. INTRODUCTION 13

2. The web server retrieves the requested document in HyperText Markup Language

(HTML) format. See Figure 1.3.2.

Figure 1.3.2.

3. The web server serves the HTML document to the web browser. And the web browser

renders the HTML to the users. See Figure 1.3.3.

Figure 1.3.3.

1.3.2 The Modern World Wide Web

The modern World Wide Web can be viewed through the model-view-controller (MVC)

model. The steps are as follows:

1. INTRODUCTION 14

1. As before, the web browser requests information from the web server via a URL.

This time the requested document may represent an online application rather than

just a static HTML ﬁle. Fee Figure 1.3.4.

Figure 1.3.4.

2. The web server loads the online application, which then determines the response.

The application may request information from a back-end database. See Figure 1.3.5.

Figure 1.3.5.

1. INTRODUCTION 15

3. The web server serves the response to the web browser. However, the response may

come in XML format rather just HTML format. See Figure 1.3.6.

Figure 1.3.6.

1.3.3 The Semantic Web

The modern architecture described above can support the Semantic Web. The enhance-

ment of the Semantic Web case is the incorporation of Resource Description Framework

(RDF) ﬁles in the process. To be more speciﬁc, the response that the application produces

is in RDF format, which is a form of XML. Details about how information is stored in the

Semantic Web and how RDF ﬁle is formulated will be discussed in Chapter 2.

Sharing mathematics information on the Semantic Web can overcome the drawbacks of

the previously mentioned sharing methods such as the lack of details, accuracy, and ver-

satility. This project will focus on representing and publishing ﬁnite algebras, which is a

narrow ﬁeld in mathematics, on the Semantic Web.

Triples and the Semantic Web

The Semantic Web, which serves as a massive, globally-linked, distributed database, allows

users to store and query information from the World Wide Web. In order to generalize

the Semantic Web, those who enter information into this database will have to adhere to

a common standard syntax.

2.1 Triples

Information in the Semantic Web is decomposed and stored into small pieces, called

triples[1]. A triple consists of a subject, a predicate and an object. The subject denotes the

resource, and the object denotes the target. The predicate denotes traits or aspects of the

resource and expresses a relationship between the subject and the object. For example,

given a simple statement:

“I own my apartment.”

Figure 2.1.1.

We can decompose it as:

2. TRIPLES AND THE SEMANTIC WEB 17

Figure 2.1.2.

If we complicate the statement in Figure 2.1.1 and turn it to:

“I own my apartment, which has a balcony.”

Figure 2.1.3.

We can enhance the relation in Figure 2.1.2 to:

Figure 2.1.4.

As we can see, there are two triples in Figure 2.1.4. The ﬁrst one is what we have in

Figure 2.1.4. It has “I” as the subject, “my apartment” as the object and “own” as the

predicate to indicate the relation between “I” and “my apartment”. The second triple

has “my apartment” as the subject, “has” as the predicate and “balcony” as the object.

Notice that these two triples share a piece of information, which is “my apartment”. The

2. TRIPLES AND THE SEMANTIC WEB 18

“my apartment” “I” “own” is the same “my apartment” that “has” “balcony”. So “my

apartment” is both the object of the ﬁrst triple and the subject of the second triple. Thus,

in Figure 2.1.4, the “my apartment” unit points to both subject and object tags. If we have

a table storing the information such as the table in Figure 2.1.5, we are able to retrieve

the complete information of the statement in Figure 2.1.3.

Subject Predicate Object

I own my apartment

my apartment has balcony

Figure 2.1.5.

If we keep complicating the statement in Figure 2.1.3, we can update the table in Figure

2.1.5 by appending more triples that capture the new information. In fact, if we think

in an opposite direction, any complex statements can be decomposed to a collection of

simple statements. The examples we mentioned above are naturally subject-verb-object

structured, so it is obvious to see how we can reduce it. For statements that are not natu-

rally subject-verb-object structured, the approach is not obvious. For example, let us look

at the following statement:

“Bob gives a computer to Lily.”

Figure 2.1.6.

In the above statement, we have “Bob” as the subject, “gives” as the verb, “a computer”

as the object and “to Lily” as the indirect object. Since indirect object is involved here, it

is not likely to translate it directly to triples without losing information. The solution is to

give the statement an identiﬁer and take it as the subject in triples. Then for each segment

of the statement, we take the type of the segment as the predicate and the content of the

segment as the object. So let us name the statement in Figure 2.1.6 as SEN1. Then we

can store the information of SEN1 as below:

2. TRIPLES AND THE SEMANTIC WEB 19

Subject Predicate Object

SEN1 subject Bob

SEN1 verb gives

SEN1 object a computer

SEN1 indirect object to Lily

Figure 2.1.7.

In this way, we manage to decompose a non-subject-verb-object-structured statement to

a list of triples. Thus, we can transform any single arbitrary statement to a list of triples

while the information of that statement is preserved within the triples. If we have a col-

lection of statements, we can still get the list of triples of the complete information by

joining each list of triples from each statement. Therefore, the claim is that any database

can be reduced to a collection of triples.

So let us keep complicating the statement in Figure 2.1.3:

“I own my apartment, which has a balcony. My friend Susie also owns her apartment,

which also has a balcony.”

Figure 2.1.8.

The graphical presentation of the above statement in Figure 2.1.9. Here we represent the

Figure 2.1.9.

relation diﬀerently from Figure 2.1.2 and Figure 2.1.4. In Figure 2.1.9, subject and object

2. TRIPLES AND THE SEMANTIC WEB 20

units are denoted by rectangles and predicate units by ellipses. An arrow indicates a triple

relation. The beginning of the arrow connects to the subject unit of the triple relation and

the end of the arrow points to the object unit. The predicate unit of the triple relation is

attached to the middle of the arrow. In Figure 2.1.9, we have ﬁve triples.

Then we run into a problem. In Figure 2.1.9, there are two object units that are both

named “balcony” but refer to two diﬀerent object. If we encounter a “balcony” object

when we are retrieving the information form the triples, we would not be able to tell

which “balcony” it really points to. Similarly, having two units named “my apartment”

and “her apartment” would be vague and problematic. The solution is to number the same

type of things in order, so that we don’t get confused when we retrieve information back.

The updated graphical representation is as below:

Figure 2.1.10.

Figure 2.1.10 will ﬁx the problem we mentioned above. However, Figure 2.1.10 is still prob-

lematic. I name my apartment as “apartment1” and Susie’s apartment as “apartment2” so

that in my local database, these two apartments can be distinguished. However, someone

else might call his apartment “apartment1” in his local database. There would be no con-

fusion if both of our databases stay disconnected. But if both of us submit our databases

to the Semantic Web, then we would run into the problem of ambiguity again. To ﬁx this

problem completely, we replace all the names we deﬁne to Uniform Resource Identiﬁers.

2. TRIPLES AND THE SEMANTIC WEB 21

Uniform Resource Identiﬁer, in short URI, is the generic term for all types of names and

addresses that refer to resources on the World Wide Web. It is consisted of a string of

characters. Below are a few examples of URIs:

• http://example.org/stuﬀ/1.0/homePage

• ftp://example.org/resource.txt

• http://www.asclab.org/zheng

Speciﬁcally, an URI starts with a protocol “http://”, followed by the server name, a unix

path name, and an optional string representing a unix ﬁlename or application.

Thus, we can modify the triples in Figure 2.1.10 to:

Subject Predicate Object

http://www.asclab.org/zheng/example/ name “my name”

-person/1

http://www.asclab.org/zheng/example/ own http://www.asclab.org/zheng/example/

-person/1 -apartment/1

http://www.asclab.org/zheng/example/ friend with http://www.asclab.org/zheng/example/

-person/1 -person/2

http://www.asclab.org/zheng/example/ has http://www.asclab.org/zheng/example/

-apartment/1 -balcony/1

http://www.asclab.org/zheng/example/ name “Susie”

-person/2

http://www.asclab.org/zheng/example/ own http://www.asclab.org/zheng/example/

-person/2 -apartment/2

http://www.asclab.org/zheng/example/ has http://www.asclab.org/zheng/example/

-apartment/2 -balcony/2

Figure 2.1.11.

With URIs, we would no longer have the problem of ambiguity because every item is

represented by a unique URI.

2. TRIPLES AND THE SEMANTIC WEB 22

2.2 Representing Triples

In the following section, we will introduce two diﬀerent representations of triples and

discuss the features of each representation.

2.2.1 Resource Description Framework (RDF)

Resource Description Framework (RDF)[9] is the standard framework for encoding meta-

data and describing other resources on the Semantic Web. RDF is an abstract model, a

way to break down knowledge into discrete pieces. While being designed to be read and

understood by computers, it is not meant to be displayed to people. RDF identiﬁes things

using URIs, which we have discussed the advantage of doing so. The common format

of storing RDF is XML. It describes resources(subjects) with properties(predicates) and

property values(objects) in the following basic structure:

<?xml version=“1”? >

<RDF>

<Description about=“subject

< predicate1 > object1 < /predicate1 >

< predicate2 > object2 < /predicate2 >

< /Description>

< /RDF>

Figure 2.2.1.

Notice that all the triples with the same subject are included under the same parent “De-

scription” tag. Each triple forms a child tag with the predicate as the tag name and the

object as the content of the tag. If we have multiple subjects, we append more “Descrip-

tion” tags under the root “RDF” tag.

The triples in Figure 2.1.11 would look like the following in RDF format:

2. TRIPLES AND THE SEMANTIC WEB 23

<?xml version=“1”? >

<RDF>

<own>http://www.asclab.org/zheng/example/apartment/1< /own>

<friend with>http://www.asclab.org/zheng/example/person/2< /friend with>

< /Description>

<has>http://www.asclab.org/zheng/example/balcony/1< /has>

< /Description>

<name>“Susie”< /name>

<own>http://www.asclab.org/zheng/example/apartment/2< /own>

< /Description>

<has>http://www.asclab.org/zheng/example/balcony/2< /has>

< /Description>

< /RDF>

Figure 2.2.2.

RDF format can store every detail information of triples. So when RDF ﬁles are passed to

computers, those computers obtain the full-scale database that they can retrieve informa-

tion from. However, the RDF-XML rules of structuring and tedious syntax make it very

hard for people to read and understand the relations between resources.

2.2.2 Terse RDF Triple Language (Turtle)

Terse RDF Triple Language (Turtle)[8] is a plain-text RDF representation, which is rep-

resenting information using “triples”. Each triple has the basic format:

< subject >< predicate >< object > .

Figure 2.2.3.

Each statement ends with a period. The subject element and the predicate element in

the triple are URIs. The object element, however, can be represented in a form of string,

number or URI. “,” is introduced for statements with the same subject and predicate.

2. TRIPLES AND THE SEMANTIC WEB 24

we can make the document shorter and more readable by combining the statements and

separating the objects by one or more commas.

< subject >< predicate >< object1 >, < object2 >, < object3 > .

Figure 2.2.4.

“;” is introduced for statements with the same subject but diﬀerent predicates and objects.

We combine them and separate the predicate-object parts with a semicolon.

< subject >

< predicate1 >< object1 >;

< predicate2 >< object2 > .

Figure 2.2.5.

Besides the basic format, preﬁxes are allowed by adding following string to the top of the

Turtle ﬁle:

@preﬁx pref: < URI pr ef ix > .

Figure 2.2.6.

Turtle also allows us to deﬁne the type relationship, which is a relationship between a

thing and a category of things, by a special keyword a.

< subject > a type name.

Figure 2.2.7.

When we don’t want to deﬁne an URL for something, we can use an anonymous object

in the object area in Turtle. Anonymous objects are deﬁned with square brackets “[ ]”

around one or more predicate-object pairs, deﬁning the object, separated by semicolons.

We can deﬁne all kinds of relationships that that object is the subject of.

< subject1 >< predicate1 > [< predicate2 >< object2 >; < predicate3 >< object3 >].

Figure 2.2.8.

2. TRIPLES AND THE SEMANTIC WEB 25

The triples in Figure 2.1.11 should be presented as following in the Turtle format:

@preﬁx : <http://www.asclab.org/zheng/example/>.

@preﬁx person: <http://www.asclab.org/zheng/example/person/>.

@preﬁx apartment: <http://www.asclab.org/zheng/example/apartment/>.

@preﬁx balcony: <http://www.asclab.org/zheng/example/balcony/>.

person:1

:name “my name”;

:own apartment:1;

:friend with person:2.

apartment:1 :has balcony:1.

person:2

:name “Susie”;

:own apartment:2.

apartment:2 :has balcony:2.

Figure 2.2.9.

Figure 2.2.9 can be further simpliﬁed as below:

@preﬁx : <http://www.asclab.org/zheng/example/>.

@preﬁx person: <http://www.asclab.org/zheng/example/person/>.

@preﬁx balcony: <http://www.asclab.org/zheng/example/balcony/>.

person:1

:name “my name”;

:own [:has balcony:1];

:friend with [:name “Susie”; :own [:has balcony:2]].

Figure 2.2.10.

As we can see, Turtle format is good at presenting resource information to readers. The

syntax is more concise and readable. For example, the representation in Figure 2.2.9 is

much more straightforward and pleasant to look at than the representation in Figure 2.2.2.

It also allows us to reconstruct its structure so the relationships among diﬀerent resources

are more obvious and focused to readers. For example, Figure 2.2.10 gives readers a better

2. TRIPLES AND THE SEMANTIC WEB 26

understanding of how each resource is related to “person:1”.

As for the downside of Turtle format, the rules of structuring a Turtle ﬁle is more complex.

Also, some information might be excluded while we are simplifying the ﬁle. For example,

we lost the information of “apartment:1”, “apartment:2” and “person:2” in Figure 2.2.10.

We only know “person:1” “own” somethings that “has” “balcony:1” for instance.

Quandles

This chapter deﬁnes algebras and quandles in mathematics. It also introduces some of the

research projects concerning quandles, which shows the need of sharing quandles and their

properties on the Semantic Web.

3.1 Algebra

3.1.1 General Deﬁnitions and Axioms

Deﬁnition. An operation f on a set s is a function f : S × S × · · · × S −→ S.

∗ is said to be a binary operation if ∗ is a function:

∗ : S × S −→ A.

And A is said to be closed under the binary operation ∗. Usually we denote ∗(a, b) by

a ∗ b where a, b ∈ S. We say ∗ is associative if

(a ∗ b) ∗ c = a ∗ (b ∗ c)

for all a, b, c ∈ S. And we say ∗ is commutative if

a ∗ b = b ∗ a

3. QUANDLES 28

for all a, b, c ∈ S. If there exists an element e ∈ S such that

e ∗ a = a = a ∗ e

for all a ∈ S, then e is called the identity. If for each a ∈ S, there exists a

−1

∈ S such

that

a ∗ a

−1

= e = a

−1

∗ a,

then a

−1

is called the inverse of a.

Deﬁnition. An ﬁnite algebra A is a pair (S, F), where S is a ﬁnite set and F is a

set of ﬁnitary operations on S.

3.1.2 Example of Algebras

One of the common types of algebra is group.

Deﬁnition. A group G = (S, ·, ()

−1

, e) is a set S with a binary operation · on A if the

following axioms holds:

• (Associativity:) For all a, b, c ∈ S, (a · b) · c = a · (b · c).

• (Identity:) There exists an element e ∈ S with e · a = a = a · e for all a ∈ S.

• (Inverse:) For each a ∈ S, there exists a

−1

∈ S such that a

−1

· a = e = a · a

−1

One example of a group is G = (I, +, (−)

−1

, 0), where

• I is the set of integers.

• + is the operation of addition.

• 0 is the identity.

• the inverse operation is negation.

3. QUANDLES 29

3.2 Quandles

3.2.1 Deﬁnition

Deﬁnition. A quandle Q = (S, ∗, /) is a ﬁnite algebra, containing two binary operations

that satisﬁes idempotence, right cancellation and right self-distributivity on S. In another

word, for any arbitrary a, b, c ∈ A,

• (Idempotence:) a ∗ a = a.

• (Right Cancellation:) (a ∗ b)/b = a and (a/b) ∗ b = a

• (Right Self-Distributivity:) (a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c).

3.2.2 Examples

A simple example of a quandle is the Unary Quandle on two elements U

= ({0, 1}, ∗, /),

where

∗ 0 1

0 0 0

1 1 1

and

/ 0 1

0 0 0

1 1 1

According to the two unary operation table, the following equations hold:

0 ∗ 0 = 0

0 ∗ 1 = 0

1 ∗ 0 = 1

1 ∗ 1 = 1

0 / 0 = 0

0 / 1 = 0

1 / 0 = 1

1 / 1 = 1

Another example of a quandle is the Tait Quandle ({0, 1, 2}, ∗, /), where

∗ 0 1 2

0 0 2 1

1 2 1 0

2 1 0 2

and

/ 0 1 2

0 0 2 1

1 2 1 0

2 1 0 2

Notice that both example quandles we have shown above have the same ∗ and / bi-

nary operations. However, this is not common among most quandles that have been dis-

3. QUANDLES 30

covered. Most quandles have very diﬀerent ∗ and / binary operations. An example is

({0, 1, 2, 3, 4}, ∗, /), where

∗ 0 1 2 3 4

0 0 2 3 4 1

1 2 1 4 0 3

2 3 4 2 1 0

3 4 0 1 3 2

4 1 3 0 2 4

and

/ 0 1 2 3 4

0 0 3 4 1 2

1 4 1 3 2 0

2 1 0 2 4 3

3 2 4 0 3 1

4 3 2 1 0 4

3.2.3 Results from the ASC Lab

The Bard College Laboratory for Algebraic and Symbolic Computation (ASC) has been

conducting research concerning the theory quandles over the years. Below is a list of ASC

lab research topics:

• Term Rewriting System[6]

The lab discovered a term rewriting system, which is both conﬂuent and terminating,

for the ﬁrst-order equational theory of quandles. This system allows one to compute

unique normal forms for quandle expressions. Moreover, the system can be used to

decide whether a given equation is also an identity for the theory. Related problems

such as the Towers of Hanoi and related algebras such as racks are also in the mix.

Here are some mathematics objects that we are interested in storing:

– Quandles expressed as tables;

– Rewriting rules expressed as pairs of expressions; and

– Normal forms expressed as expressions.

• Word Problem[5]

The Word Problem project reduced the word problem for recursively presented

groups to the word problem for recursively presented quandles. It proves that, in

general, the word problem for recursively presented quandles is not eﬀectively com-

putable. The feature of not-eﬀectively-computable also extends to the word problem

3. QUANDLES 31

for recursively presented racks as this project has also demonstrated.

Here are some mathematics objects that we are interested in storing:

– Finitely presented quandles expressed as tables;

– Groups expressed as expressions;

– Group quandles expressed as expressions; and

– Racks expressed as tables.

• Constraint Satisfaction Problem (CSP)[4]

The lab has also considered the relation between constrain satisfaction problems

and ﬁnite quandles. In particular, they proved that the tractability of a quandle

is related to the connectivity of its right Cayley graph. It turns out that a strong

notion of connectivity in the Cayley graph governs CSP dichotomy in the class of

ﬁnite quandles.

Here are some mathematics objects that we are interested in storing:

– Cayley graphs expressed as graphs;

– Malcev terms expressed as expressions; and

– Merling terms expressed as expressions.

• QuandleViewer

The QuandleViewer project has developed a graphic user interface written in Math-

ematica. It allows users to view the inputed quandles expressed as operation ta-

bles and Cayley graphs. The program takes the Mace4 portable format (.P), which

contains information of quandles, as an input, and translates it to a Mace4-style

Mathematica expression, which is called QuandleViewer format.

Quandles and Triples Transformation

4.1 Breaking Down A Quandle

Now we will show how to present a quandle as a list of triple lists in Mathematica. We

will start with representing an operation table in the “list of lists” Mathematica format.

Then we will show how to constitute the “list of lists” of a quandle.

4.1.1 From operation table to “list of lists” Mathematica format

An example of an operation table O is as below:

∗ 0 1 2

0 0 2 1

1 2 1 0

2 1 0 2

This is a binary operation with ∗ as the operation symbol. An operation table contains a

column of size n, a header of size n and n rows of size n. When applying the operation,

we choose the ith element from the column as a and the jth element from the header as

b. Then a ∗ b equals the jth element from the ith row of the operation table. Using the

operation table O, for example, we have the following equations:

4. QUANDLES AND TRIPLES TRANSFORMATION 33

0 ∗ 0 = 0

0 ∗ 1 = 2

0 ∗ 2 = 1

1 ∗ 0 = 2

1 ∗ 1 = 1

1 ∗ 2 = 0

2 ∗ 0 = 1

2 ∗ 1 = 0

2 ∗ 2 = 2

We can break O into six parts:

• ∗: the operation type at the upper left.

•

0 1 2

: the row header RH at the upper right.

•

: the column header CH at the lower left.

•

0 2 1

: the ﬁrst row R1 from the lower right table.

•

2 1 0

: the second row R2 from the lower right table.

•

1 0 2

: the third row R3 from the lower right table.

Thus, O can be described with six triples:

• {O, operation type, ∗ }

• {O, row header, RH }

• {O, column header, CH }

• {O, ﬁrst row, R1 }

• {O, second row, R2 }

• {O, third row, R3 }

4. QUANDLES AND TRIPLES TRANSFORMATION 34

We can put these four triples into a list as below and form a list of list:

{ {O, operation type, ∗},

{O, row header, RH},

{O, column header, CH},

{O, ﬁrst row, R1},

{O, second row, R2},

{O, third row, R3} }.

Figure 4.1.1.

Notice that some of the predicates contains ranking information such as “ﬁrst row”, “sec-

ond row” and “third row”. This is not appropriate. In order to avoid that, we introduce

three intermediate identiﬁers RT 1, RT 2 and RT 3. R T 1 refers to the ﬁrst row of O and

R1 refer to the row

0 2 1

. Then we can turn the triple {O, ﬁrst row, R1} into three

triples in Figure 4.1.2 in which none of the predicates contains the ranking information.

{ {O, row, RT 1},

{RT 1, place, 1},

{RT 1, value, R1} },

Figure 4.1.2.

We can apply the same method to the second and the third row of O and replace with

{O, second row, R2 } and {O, third row, R3 } in the list in Figure 4.1.1. Thus, we can

turn the list in Figure 4.1.1 to the list in Figure 4.1.3.

At this point, we still need to add more information of RH, CH, R1, R2 and R3 to the

triple list. Let us look at RH ﬁrst. RH can be break into 3 parts:

{ {RH, ﬁrst element, 0},

{RH, second element, 1},

{RH, third element, 2} }.

Here the predicates contains ranking information. We can solve this problem by using the

same approach we had for the rows. Then we can describe R H as below:

4. QUANDLES AND TRIPLES TRANSFORMATION 35

{ {O, operation type, ∗},

{O, header, RH},

{O, column, CH},

{O, row, RT 1},

{RT 1, place, 1},

{RT 1, value, R1},

{O, row, RT 2},

{RT 2, place, 2},

{RT 2, value, R2},

{O, row, RT 3},

{RT 3, place, 3},

{RT 3, value, R3} }.

Figure 4.1.3.

{ {RH, element, ERH1},

{ERH1, place, 1},

{ERH1, value, 0},

{RH, element, ERH2},

{ERH2, place, 2},

{ERH2, value, 1},

{RH, element, ERH3},

{ERH3, place, 3},

{ERH3, value, 2} }

Figure 4.1.4.

Then we can include the detailed information of H by joining the list in Figure 4.1.4

to the list in Figure 4.1.3. Notice that, similar to H, C, R1, R2 and R3 are all lists with

a sequence of numbers. Thus, we can handle them in the same way that we handle H.

Therefore, we can derive a list with the complete information in the end. The complete list

is shown as in Figure 4.1.6. In Mathematica, the list of lists would have the representation

as suggested below:

Figure 4.1.5.

4. QUANDLES AND TRIPLES TRANSFORMATION 36

Figure 4.1.6.

4. QUANDLES AND TRIPLES TRANSFORMATION 37

4.1.2 From quandle to “list of lists” Mathematica format

By deﬁnition, A quandle Q = (S, ∗, /) contains a set S and two binary operations ∗, /.

Each operation can be represented using a operation table. For example, the Tait Quandle

T Q = ({0, 1, 2}, ∗, /), where

∗ 0 1 2

0 0 2 1

1 2 1 0

2 1 0 2

/ 0 1 2

0 0 2 1

1 2 1 0

2 1 0 2

We can break T Q into 3 parts:

• {0, 1, 2}: the set S

•

∗ 0 1 2

0 0 2 1

1 2 1 0

2 1 0 2

: the ∗ operation OP 1

•

/ 0 1 2

0 0 2 1

1 2 1 0

2 1 0 2

: the / operation OP 2

Thus, the triple representation is:

{ {T Q, set, S},

{T Q, ﬁrst operation, OP 1},

{T Q, second operation, OP 2} }

Figure 4.1.7.

The list in Figure 4.1.7 can then be modiﬁed to:

{ {T Q, set, S},

{T Q, operation, OP T Q1},

{OP T Q1, place, 1},

{OP T Q1, value, OP 1},

{T Q, operation, OP T Q2},

{OP T Q2, place, 2},

{OP T Q2, value, OP 2} },

Figure 4.1.8.

4. QUANDLES AND TRIPLES TRANSFORMATION 38

in order to avoid including the ranking information in the predicates. Then we need to ﬁll

in the information of S, OP 1 and OP 2 by appending more triple lists to the list in Figure

4.1.8.

Notice that OP1 is same as the example operation table we showed earlier. Thus, we can

describe it as we did in Figure 4.1.6. The only change we are making is replacing O with

OP 1 because we refer to the operation table as OP 1 now. OP 2 shares the same header,

column and rows as OP 1 except for the operation type. Thus, we can add the information

of OP2 by appending these triple lists to the list in Figure 4.1.6:

{ {OP 2, operation type, /},

{OP 2, header, RH},

{OP 2, column, CH},

{OP 2, row, RT T 1},

{RT T 1, place, 1},

{RT T 1, value, R1},

{OP 2, row, RT T 2},

{RT T 2, place, 2},

{RT T 2, value, R2},

{OP 2, row, RT T 3},

{RT T 3, place, 3},

{RT T 3, value, R3} }.

Figure 4.1.9.

Notice here we do not need to include the information of RH, CH, R1, R2 and R3 any

more because this information is already existed in the list in Figure 4.1.6. Joining the

lists in Figure 4.1.6 and 4.1.9 would give us the complete information of the two binary

operations OP 1 and OP 2. The lists are shown in Figure 4.1.10.

4. QUANDLES AND TRIPLES TRANSFORMATION 39

Figure 4.1.10.

4. QUANDLES AND TRIPLES TRANSFORMATION 40

The last thing we need to include from the Tait Quandle is the set S. We can describe

S within 3 more triple lists:

{ {S, element, 0},

{S, element, 1},

{S, element, 2} }

Figure 4.1.11.

Hence, by joining the lists in Figure 4.1.8, 4.1.10 and 4.1.11, we would get the list that

contains the complete information of the Tait Quandle in Figure 4.1.12

4. QUANDLES AND TRIPLES TRANSFORMATION 41

Figure 4.1.12.

4. QUANDLES AND TRIPLES TRANSFORMATION 42

4.2 Transformation Implementation

4.2.1 From Mathematica list of triple lists to RDF

In Mathematica[3], quandle information can be stored as a list of triple lists as above.

The input format is the a list of triples in the Mathematica list format:

Figure 4.2.1.

As we can see, this format contains two levels of lists.

• First level: each individual triple list;

• Second level: the collection of all triples.

To translate the triples in Mathematica list format to the RDF format, we need to ﬁrst

change the structure of the list and make it a list of three levels as below:

Figure 4.2.2.

4. QUANDLES AND TRIPLES TRANSFORMATION 43

The new list contains three levels:

• First level: each individual triple list;

• Second level: the collection of all triples that contains the same subject, which is the

ﬁrst element of an individual triple list;

• Third level is the collection of all the second-level lists.

The translation of the two-level list to the three-level list could be done by calling

GatherBy[ , First] in Mathematica. This will also sort each secondary list according to

the second element of each individual triple. The new list format would be helpful for the

actual translation from the Mathematica triple in list format to the RDF format.

With the three-level list,

1. We create an empty list called RDF ElementList.

2. We loop through all the secondary lists. For each secondary list, we create a tempo-

rary empty list called tmpRDFElementList.

3. Then we loop through all the triple lists in the secondary list. For each individual

triple list, we create an XML tag with the predicate, which is the second element

of the triple list, as the tag name and the object, which is the third element of the

triple list, as the content of the “rdf:resource” attribute of the tag.

4. We append the XML tags we create for each individual triple list to tmpRDF E lementList.

Step 2 through 4 are implemented in the code in Figure 4.2.3.

5. We then return to the secondary level. We obtain the subject, which is the ﬁrst

element of an individual triple list. Since all the triple lists inside a secondary list

contains the same subject, it does not matter which individual triple list is chosen.

4. QUANDLES AND TRIPLES TRANSFORMATION 44

Figure 4.2.3.

With the subject value we then create an XML tag with “rdf:Description” as the

tag name, the subject value as the content of the “rdf:about” attribute of the tag

and the temporary list we have above as the content of the tag.

6. We append the XML tag we create for each secondary list to RDF El ementList.

Step 5 through 6 are implemented in the code in Figure 4.2.4.

Figure 4.2.4.

The resulting RDF representation from Figure 4.2.1 is shown in Figure 4.2.5.

To complete the RDF ﬁle, the last thing we need to do is adding a parent XML tag around

the XML tags in Figure 4.2.5.

Figure 4.2.6.

4. QUANDLES AND TRIPLES TRANSFORMATION 45

Figure 4.2.5.

The complete RDF ﬁle would have the representation as below:

Figure 4.2.7.

Conclusion

This project looks into the problem of storing and sharing mathematics information on the

World Wide Web. In particular, we consider the encoding of ﬁnite algebras from universal

algebras as triples for the Semantic Web. The project focuses on the collection of ﬁnite

quandles, which is a class of ﬁnite algebras.

Included are various pieces of software that have been developed to translate between

diﬀerent formats for expressing quandles. The target formats involved are listed as below:

• Mace4 Portable Format (.P). See example in Figure 5.0.1.

interpretation( 3, [number = 5,seconds = 0], [

function(∗( , ), [0,2,1,2,1,0,1,0,2]),

function(/( , ), [0,2,1,2,1,0,1,0,2])]).

Figure 5.0.1.

• QuandleViewer Format (Mace4-style Mathematica Expression). See example in Fig-

ure 5.0.2.

5. CONCLUSION 47

interpretation[3, 5, 0, {

function[∗, 2, {{1, 3, 2}, {3, 2, 1}, {2, 1, 3}}],

function[/, 2, {{1, 3, 2}, {3, 2, 1}, {2, 1, 3}}]}]

Figure 5.0.2.

• Prolog-style Triples. See example in Figure 5.0.3.

rdf(‘I’, ‘own’, ‘my apartment’).

Figure 5.0.3.

• RDF-XML

Moreover, we have the following intermediate formats to simplify the translation functions:

• Mathematica List of Mathematica-style Triples

• Mathematica List of List of Triple-link

• Mathematica List of Prolog-style Triple Strings

The translation relations are shown in the Figure 5.0.4.

The work that has been done in this project sets a foundation for representing all levels

of properties of quandles in the Semantic Web. To be able to store quandle objects in the

Semantic Web allows us to append information that states certain properties and relations

about the quandle objects in the linked database.

5.1 Future Work

Natural projects that can continue from this project include the following:

• Storing some representations of expression such as equations, presentations, and

rewriting systems;

5. CONCLUSION 48

Figure 5.0.4.

• Storing ﬁrst-order properties of quandles;

• Storing second-order properties of quandles;

• Storing other algebra objects;

• Storing information of relationships between diﬀerent kinds of algebras;

• Develop software that can search and retrieve quandle information from the Semantic

Web; and

• Develop a trust system that store proofs in the Semantic Web style and does veriﬁ-

cation of the proofs.

Bibliography

[1] Toby Segaran, Colin Evans, and Jamie Taylor, Programming the Semantic Web,

O’Reilly Media, Sebastopol, CA, 2009.

[2] Tim Berners-Lee and Mark Fischetti, Weaving the Web, HarperSanFrancisco, Chapter

12, 1999.

[3] Stephen Wolfram, The Mathematica Book, Wolfram Media, Incorporated, 2003.

[4] Brita Brudvig, Benjamin Fish, Solomon Garber, Max Jeter, Liwen Song, and Bob

McGrail, CSPs and Connectedness: P/NP-complete Dichotomy for Idempotent, Right

Quasigroups (2012).

[5] James Belk and Bob McGrail, The General Word Problem for Quandles is Undecid-

able (2012).

[6] Peter Golbus, Claudio Gutierrez, Bob McGrail, and Hannah Vallee, Quandles, Term

Rewriting Systems, and the Towers of Hanoi (2011).

[7] James Hendler, Tim Berners-Lee, and Eric Miller, Integrating Applications on the

Semantic Web, Journal of the Institute of Electrical Engineers of Japan Vol 122(10)

(2002), 676–680.

[8] W3C Team Submission. Turtle - Terse RDF Triple Language, http://www.w3.org/

TeamSubmission/turtle/. Accessed (April 24, 2013).

[9] W3C Recommendation. RDF/XML Syntax Speciﬁcation (Revised), http://www.w3.

org/TR/rdf-syntax-grammar/. Accessed (April 24, 2013).

[10] The 9th International Conference on Mathematical Knowledge Management, http:

//cicm2010.cnam.fr/mkm/. Accessed (April 25, 2013).

[11] DML 2010, 3nd workshop/conference: Towards a Digital Mathematics Library, http:

//www.fi.muni.cz/

sojka/dml-2010.html\#topics. Accessed (April 25, 2013).

[12] OpenMath Website, http://www.openmath.org/index.html. Accessed (April 25,

2013).

Bibliography 50

[13] American Mathematical Society (AMS), http://www.ams.org/home/page. Accessed

(April 28, 2013).

[14] UACalc, A Universal Algebra Calculator, http://uacalc.org/. Accessed (April 28,

2013).

[15] The KnotPlot Site, http://www.knotplot.com/. Accessed (April 28, 2013).

[16] Prover9 and Mace4, http://www.cs.unm.edu/

mccune/mace4/. Accessed (April 28,

2013).

[17] Wolfram Mathematica: Technical Computing Software, http://www.wolfram.com/

mathematica/. Accessed (April 28, 2013).

[18] Maple 17 - Technical Computing Software for Engineers, http://www.maplesoft.

com/products/maple/. Accessed (April 28, 2013).

[19] MATLAB - The Language of Technical Computing, http://www.mathworks.com/

products/matlab/. Accessed (April 28, 2013).

[20] Wolfram—Alpha: Computational Knowledge Engine, http://www.wolframalpha.

com/. Accessed (April 28, 2013).

[21] Wolfram MathWorld: The Web’s Most Extensive Mathematics Resource, http://

mathworld.wolfram.com/. Accessed (April 28, 2013).

[22] Knot Atlas: Main Page, http://katlas.math.toronto.edu/wiki/Main_Page. Ac-

cessed (April 28, 2013).

[23] Wikipedia, http://www.wikipedia.org/. Accessed (April 28, 2013).